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Nov 17 Multivariable Second Derivative Test

Multivariable Second Derivative Test

In single variable calculus, we used the second derivative test as an alternative to doing the first derivative test to determine minima/maxima. Now, this is the multivariable version.

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$$\text{Let } g(x,y) = f_{xx}(x,y)f_{yy}(x,y) - \left(f_{xy}(x,y)\right)^2$$
This occurs at a critical point, because we know this could be an extremum, we just want to know which kind. So $f_x = 0$ and $f_y = 0$